Problem: Is ${666652}$ divisible by $3$ ?
A number is divisible by $3$ if the sum of its digits is divisible by $3$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {666652}= &&{6}\cdot100000+ \\&&{6}\cdot10000+ \\&&{6}\cdot1000+ \\&&{6}\cdot100+ \\&&{5}\cdot10+ \\&&{2}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {666652}= &&{6}(99999+1)+ \\&&{6}(9999+1)+ \\&&{6}(999+1)+ \\&&{6}(99+1)+ \\&&{5}(9+1)+ \\&&{2} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {666652}= &&\gray{6\cdot99999}+ \\&&\gray{6\cdot9999}+ \\&&\gray{6\cdot999}+ \\&&\gray{6\cdot99}+ \\&&\gray{5\cdot9}+ \\&& {6}+{6}+{6}+{6}+{5}+{2} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $3$ , so the first five terms must all be multiples of $3$ That means that to figure out whether the original number is divisible by $3 $ , all we need to do is add up the digits and see if the sum is divisible by $3$ . In other words, ${666652}$ is divisible by $3$ if ${ 6}+{6}+{6}+{6}+{5}+{2}$ is divisible by $3$ Add the digits of ${666652}$ $ {6}+{6}+{6}+{6}+{5}+{2} = {31} $ If ${31}$ is divisible by $3$ , then ${666652}$ must also be divisible by $3$ ${31}$ is not divisible by $3$, therefore ${666652}$ must not be divisible by $3$.